Reflection Method for Mathematical Modeling of Potential Fields in Multi-Layer Media

In this paper, potential fields in areas with plane and circular symmetry have been studied. In this case, the field potential is defined as the sum of solutions of Dirichlet model boundary value problems. The reflection method is used for the modeling of stationary thermal fields in multilayer media. By applying the reflection method, we found analytical solutions of boundary value problems with boundary conditions of the fourth kind for the Laplace equations and developed new computational algorithms. The developed algorithms can be easily implemented and transformed into a computer code. First of all, these algorithms implement consistent solutions of the Dirichlet problems for the model domains that allows using libraries of subroutines. Secondly, they have high algorithmic efficiency. It has been shown that reflection method is identical to the method of transformation operators and proved that transformation operator can be decomposed into a series of successive reflections from the external and internal boundaries. Finally, a physical interpretation of the reflection method has been discussed in detail.


Introduction
We note the methods of the theory of functions of a complex variable as an analytical methods for solving boundary value problems in multilayer media. Such methods lead to Riemann boundary value problems, the solution of which is accompanied by significant difficulties, for example, the calculation of the index of the boundary value problem and integrals of Cauchy type [1]. Numerical methods: finite difference method (FDM see [4]), finite element method (FEM see [2]), radial basis function method (RBFM see [3]), -also have disadvantages. For example, in the FDM-method the solution is obtained only in the grid nodes. Any area can be described with FDM, since triangles and tetrahedra easily cover even complex domains. In all the subdomains, you can easily increase the density of the computational grid to improve the accuracy of calculations. However, the general method of error estimation in FDM and RBFM is absent today.
The reflection method is one of the well-known analytical methods for solving boundary value problems. Due to the wide spread of multilayer materials, the spread of this method for boundary value problems in multilayer media becomes relevant. In our article it will be shown that the method of transformation operators previously developed by the authors (see [15,16]) allows us to present the solution of the problem as a sum of successive reflections. Also in the article it is established that the method of transformation operators and the method of reflections are identical. As a consequence of this result, we have found new analytical formulas for solving boundary value problems in multilayer media.
The article offers an interpretation of the mathematical model of multilayer media as a deformation of the model of homogeneous media. New formulas for solutions obtained in the article allowed to develop effective computational algorithms on their basis.
The reflection method was firstly proposed by Kelvin, (see [5] We choose the solution of the model problem (3) For this approximation, the second boundary condition is satisfied, but the first boundary condition is not satisfied, because Iterations continue until the required accuracy is obtained. As a result, for the approximation obtained by (m+1) -reflections from the boundary x l = , we get we obtain well-known result for solution the Dirichlet problem in the strip [5] ( )

Reflection Method in Dirichlet problem for Laplace equation in two-layer strip
Modeling of a stationary thermal field in a two-layer plate bounded in the direction of the Ox-axis leads to the problem of solving a system of separate Laplace equations with Dirichlet boundary conditions under the boundary condition of fourth kind on the line , .
x x u l y u l y u l y u l y l l The Dirichlet model problem for the strip has the form: yy xx Let the solution of the model problem (8) for the strip be known. We define the first approximation of the solution of the boundary value problem (5)- (7). The second component of the solution is given by the formula ( ) To obtain the first component of the approximation ( ) 1 1 , u x y , we perform the reflection of the model solution with respect to the line x=l. Then we obtain ( ) The first approximation satisfies the Laplace equation, the boundary condition (6) on the line x L =  , and on the line 0 x = satisfies the boundary condition ( ) ( ) The first approximation has the form ( ) Consider the following Dirichlet model problem on the strip ( ) ( ) 1 1 2 1 We use the solution ( ) 1 , u x y  of the model problem (9) to obtain the second approximation. For the first and second components of the second approximation we have expressions, respectively ( ) ( ) Considering known the solution j u  of the Dirichlet model problem for the strip (9) (10), we obtain a sequence of approximations for the solution of the problem (5)- (7). Iterations continue until the required accuracy is obtained. The process ends if the function ( ) 1 0, m u y +  satisfies the specified accuracy in a uniform norm. Remark 1. The method is transferred to the case of the number of layers greater than two. In this case, the reflections should be carried out sequentially with respect to each of the internal boundaries, moving from right to left to the boundary x=0.

Reflection Method for Dirichlet problem in two-layer half-plane
Modeling of a stationary thermal field in a two-layer plate semi-bounded in the Ox-axis direction leads to the problem of solving a system of separate Laplace equations in two-layer half-plane  [15] it is proved that the solution of the Dirichlet problem (11)-(13) has the form ( ) ( ) The computational algorithm based on the exact formulas (14) - (15) consists in replacing the sum of a series by the sum of a finite number of terms. This algorithm can be interpreted as a sequence of reflections with respect to the inner and outer boundaries. This approach allows us to find out the iterative nature of the algorithm and speed up the calculation process.
Step 1. Find the first-order approximation In this case, the first-order approximation the function ( ) 1 , u x y satisfies 1) a separate system of Laplace equations (11), 2) boundary conditions of fourth kind (13), 3) the error in the boundary condition (12) has the order (1 ) / (1 ) u χ χ − +  in the uniform metric.
Step 2. Improving the approximation. If the function ( ) , k u x y is a K-order approximation, then the (k+1)order approximation is defined by the formula, The described algorithm can be interpreted in terms of the reflection method. The algorithm extends to the case of the number of layers greater than two.

Reflection Method for increasing the number of layers in Dirichlet problem for (n+1) -layer half-plane
As a model problem, we consider the Dirichlet problem for the Laplace equation in a half-plane 0 x > with under boundary condition of fourth kind on the lines , 2,..., . . The iterative computational algorithm consists of the following steps.
Step 1. The first-order approximation is calculated by the formulas ( ) Step 2. Assuming a given k-order approximation ( ) , k j u x y , the (k+1)-order approximation is defined so that the error in the boundary condition has order 2 Thus, the k-order approximation is defined by the formulas:

Conclusion
The results given in the paper allow the extension to the case of arbitrary bounded regions 0 1 , D D with boundaries 0 1 , Γ Γ . We solve the problem of modeling a stationary thermal field in a bounded two-layer plate. We set the Dirichlet problem for a system of separate Laplace equations in a two-layer domain 0 D ( ) ( )  The computational algorithm, based on exact formulas (26)-(27), is to break the series. This approach allows us to identify the iterative nature of the algorithm and speed up the calculation process. An important advantage of the algorithm is the ability to access libraries for solving